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1

Kilka uwag o (meta)filozofii matematyki

100%

Wójtowicz K.

Zagadnienia Filozoficzne w Nauce

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2007

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issue 40

12-29

PL

The present essay deals with the problem of how to choose the correct method of doing philosophy of mathematics taking into account the importance of technical mathematical results for philosophical analysis. After a short historical introduction presenting the formation of the present mathematical paradigm, it is pointed out that the current mathematical praxis has, in principle, no connection with philosophical investigations. Two radically different approaches to philosophy of mathematics are outlined. Basing on selected examples it is argued that the correct method of doing philosophy of mathematics should take into account both technical results obtained by mathematicians (which often throw a new light on old philosophical questions) and the autonomy of philosophical method.

2

What is Diagrammatic Reasoning in Mathematics?

63%

Sochański M.

Logic and Logical Philosophy

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2018

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vol. 27

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issue 4 Special Issue on Logic, Cognition, Argumentation. Guest Editors: Mariusz Urbański, Michiel van Lambalgen and Marcin Koszowy

567-581

EN

In recent years, epistemological issues connected with the use of diagrams and visualization in mathematics have been a subject of increasing interest. In particular, it is open to dispute what role diagrams play in justifying mathematical statements. One of the issues that may appear in this context is: what is the character of reasoning that relies in some way on a diagram or visualization and in what way is it distinct from other types of reasoning in mathematics? In this paper it is proposed to distinguish between several ways of using visualization or diagrams in mathematics, each of which could be connected with a different concept of diagrammatic/visual reasoning. Main differences between those types of reasoning are also hinted at. A distinction between visual (diagrammatic) reasoning and visual (diagrammatic) thinking is also considered.

3

Idea, číslo, pravidlo

63%

Kolman V.

Filosofický časopis (Philosophical Journal)

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2009

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vol. 57

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issue 5

709-732

EN

The subject of this paper is the general question of what role mathematics, or more particularly the philosophy of mathematics, plays in the work of Wittgenstein, and, also, in philosophy generally; a question I have tackled in a more extensive form, and not always very explicitly, in the book Filosofie čísla (The Philosophy of Number). Just as there, I take as my starting point Frege’s linguistic turn, beginning with the question “what is number?”, and I develop this is in a free relation (1) to Plato’s theory of ideas with mathematical objects as the middle entities, (2) Kant’s anchoring of mathematics in the pure intuitions of space and time, and (3) Wittgenstein’s conception of mathematics as one of many language-games, stemming from the understanding of a concept as rule.

4

U zródeł współczesnego nominalizmu w filozofii matematyki

63%

Frąckowiak–Ciesielska A.

Zagadnienia Filozoficzne w Nauce

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2006

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issue 39

92-114

PL

Nominalism is one of the most important positions in the contemporary philosophy of mathematics. The goal of this article is to point out various factors that have effected the rapid development of nominalism. In particular, the figures of the Polish precursors of nominalism are presented, that is, Lesniewski, Kotarbinski and Tarski. Consequently, the article of Quine and Goodman entitled 'Steps Toward a Constructive Nominalism' is discussed. The article became the ideological declaration of nominalism and initiated reconstructionist tendencies within it. The presentation is concluded with the attempt of the modal interpretation of mathematics according to Putnam. This interpretation inspired the nominalists to enrich the existing logical apparatus with new methods and tools that are more efficient in the implementation of the strategy indicated by Quine and Godman.

5

Aksjomat multiplikatywny Russela

51%

Lewandowska K.

Zagadnienia Filozoficzne w Nauce

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2012

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issue 50

152-166

PL

We present the history of two parallel (and equivalent) discoveries: the axiom of choice and the multiplicative axiom. Firstly, we consider the origins of the formulation of the multiplicative axiom. Next, we concentrate on Russell’s attitude towards the role of this axiom, which is closely related to his philosophy of mathematics. We also highlight some diﬀerences between Russell’s and Zermelo’s propositions.

6

Kilka uwag o Tezie Churcha i Aksjomacie Hilberta

51%

Olszewski A.

Zagadnienia Filozoficzne w Nauce

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2006

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issue 38

114-126

PL

Some facts concerning Church's Thesis are first reminded, then Hilbert's Axiom of Thought is formulated. Hilbert proposed this axiom in 1905. He believed that it belongs to a domain of knowledge that is prior with respect to mathematics. An attempt is made to apply this axiom to some considerations concerning Church's Thesis

7

Metoda matematyki według B. Bolzano

51%

Dadaczyński J.

Zagadnienia Filozoficzne w Nauce

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2006

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issue 38

76-113

PL

The matter under discussion is the methodology of mathematics presented by Bernard Bolzano (1782-1848) in his early pamphlet 'Beitraege zu einer begruendeteren Darstellung der Mathematik' (Prague 1810). Bolzano built, with success, the classical axiomatic-deductive method of nonspacial and atemporal concepts (Begriffe). He abandoned the traditional custom of formulating primitive concepts of deductive theories. Bolzano opposed the traditional conviction that the axioms of mathematical theories should be clear and distinct sentences. He divided the domain of nonspacial and atemporal sentences into the subdomains of objectively provable and objectively nonprovable sentences. In his view, the axioms of mathematical (deductive) theories are only the objectively nonprovable sentences, and each of the objective nonprovable sentences is an axiom of a certain deductive theory. He postulated, at the time when only the (Euclidean) geometry was axiomatized, the axiomatization of all mathematical theories.

8

Jak přeměnit kvantitu v kvalitu? : K Hegelovu pojmu míry

51%

Kolman V.

Filosofický časopis (Philosophical Journal)

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2018

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vol. 66

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issue 3

325-348

EN

The paper deals with the phenomenon of changing quantity into quality as it is developed in Hegel’s Science of Logic and its chapter on measure. First, the relation of measure to the concept of number is analyzed, particularly with respect to the concept of real numbers and their birth from the practices of counting and measuring. After that, the measure is treated as a category of speech devoted to the application of theoretical or quantitative differences within the qualitatively given experience. As such, Hegel’s talk about the transformation of quantity into quality, as well as his talk about double negation and bad infinity, are contrasted with their interpretation in Marx’s and Engels’s political economy.

DE

Der Artikel befasst sich mit dem Phänomen des Wandels von Quantität zu Qualität, der in Hegels Wissenschaft der Logik und dort im Kapitel zur Kategorie des Maßes erörtert wird. Zunächst wird das Verhältnis des Maßes und des Zahlbegriffs analysiert, insbesondere in Bezug zum Begriff der reellen Zahl und deren Entstehung in der Praxis des Rechnens und Messens. Anschließend wird das Maß als Kategorie der Sprache erfasst, die die Anwendung theoretischer oder auch quantitativer Unterscheidung auf eine qualitativ gegebene Erfahrung betrifft. In diesem Sinne werden Hegels Ausführungen zum Wandel der Quantität in Qualität, ebenso wie seine Verwendung der doppelten Negation und der schlechten Unendlichkeit in Kontrast zur entsprechenden Interpretation in der politischen Ökonomie von Marx und Engels gesetzt.

9

Kripkenstein from the mathematical point of view: a preliminary survey

51%

Janik B., Jagielloński U.

Internetowy Magazyn Filozoficzny Hybris

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2016

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vol. 34 (3)

EN

This paper deals with the problem of the impact of Kripke’s skeptical paradox on the philosophy of mathematics. By perceiving mathematics as a huge rule-following discipline, one could argue that the Kripkean nonfactualist thesis should be adopted within the philosophy of mathematics en bloc to imply a refutation of objectivity and an enforcement of a particular view on the nature of mathematics. In this paper I will discuss this claim. According to Kripke’s skeptical solution we should reject the notion of fact and adopt the use theory of meaning that could be stated as follows: ’One understands the concepts embodied in a language to the extent that one knows how to use the language correctly.’ [Shapiro 1991, 211] [Kripke 1982]. Focusing on mathematical discourse, we should ask: what are the implications of the use theory of meaning for the philosophy of mathematics? Furthermore, is the answer to the skeptical paradox consistent with selected views in philosophy of mathematics? The supposed answer to the first question is that it demands the view that mathematics should be perceived as a strictly pragmatic discipline and the rules of mathematical discourse are mere conventions. But this is too simplistic a view and the matter at hand is far more complicated.

10

Filozofia matematyki Wacława Sierpińskiego

51%

Lewandowska K.

Zagadnienia Filozoficzne w Nauce

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2016

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issue 60

53-81

PL

The main aim of the paper was to draw attention to Wacław Sierpiński as not only a great mathematician but also a philosopher. We undertook the attempt of reconstruction of Sierpiński’s philosophy. To aim this goal we mainly based ourselves on Sierpiński’s habilitation lecture entitled The concept of correspondence in mathematics. The complementation of Sierpiński’s philosophical views were conclusions from his mathematical achievements, his scheme of research on The Axiom of Choice, and his attitude to this axiom.

11

Co nowego w filozoficznym problemie matematyczności przyrody?

51%

Sokołowski L. M.

Zagadnienia Filozoficzne w Nauce

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2015

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issue 58

63-88

EN

Actually the problem of the mathematical nature of the world is a purely philosophical one, although it lies at the very foundations of physics and mathematics (and thus is very interesting for scientists) and as such it is extremely difficult to solve; in consequence any progress in the field is slow and for many it is unconvincing. In view of that I endeavour to give a more precise meaning to the statement that the structure of the material world is determined by an independent world, that of mathematical notions. The nature of the relationship between these two worlds is a great riddle and the core of this riddle is surrounded by a dense cloud of other philosophical riddles which are closely related to it, though (it seems at present) they are independent of it. I successively peel off these surrounding problems in order to get to the very core of „the mathematicality of the matter”. First I argue that physics cannot establish whether the matter might not be subject to mathematical laws of nature, then I discuss two conceptions of the nature of the physical law, the dualistic and monistic one. It seems that independently of which conception is true, none of these helps to solve the problem. In conjunction with the famous Wigner’s article of 1960 on unreasonable effectiveness of mathematics in natural sciences I indicate that the problem concerns solely the inanimate matter and does not apply to living organisms. As a next inevitable step I discuss the view of mathematics as intellectual inquiries independent of the physical world, which nonetheless perfectly fit this world; in particular I briefly present the Einstein’s conception of forming physical laws. Finally I make comments on the problem which unavoidably appears in this context, namely of whether mathematical notions are discovered or freely created; I indicate (following A. Pelczar and others) that these two concepts do not exclude each other. After this journey through a collection of problems closely accompanying that of „the mathematicality of the matter” it turns out that we come back to the starting point and we are helplessly facing the Mystery.

12

Fizyzm Rolanda Omnésa - jedność świata matematyki i fizyki. Część I: kwantowe problemy abstrakcji

51%

Grygiel W.

Zagadnienia Filozoficzne w Nauce

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2008

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issue 43

89-102

PL

Inasmuch as mathematical platonism can be clearly matched with the radical realism, there exists a possibility to point out an approach, promoted by a French physicist, Roland Omnés, that is equivalent to the Aristotelian position of moderate realism. This standpoint denies the existence of an independent universum of mathematical entities and claims that mathematics is encoded in the laws of physics. In analogy to logicism, where mathematics is considered to be reducible to logic, Omnés' position is called by him 'physism' to stress the reducibility of mathematics to the laws of physics. The goal of Roland Omnés is to construct a common philosophy of mathematics and physics where the realities of these two disciplines converge. The first part of the analysis aims at the description and critical evaluation of physism from the point of view of the adequacy of the consistent histories interpretation of quantum mechanics to provide physical basis of the abstraction of the mathematical structures from the physical reality.

13

Podmiot matematyczny Hilberta

51%

Brożek B., Olszewski A.

Zagadnienia Filozoficzne w Nauce

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2013

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issue 53

93-132

PL

The aim of this paper is threefold. First, on the basis of Gordan’s problem and Hilbert’s basis theorem we want to say a few words about the formation of Hilbert’s philosophy of mathematics in the late nineteenth and early twentieth centuries. Second, we attempt to reconstruct Hilbert’s Program highlighting the role of reasoning which is not conducted within the axiomatic system. Third, we formulate and try to justify the claim that Hilbert’s Program assumes some metaphysics of the subject that – in general terms – is identical with Kant’s transcendental subject.

14

Matematyka eksperymentalna – kilka refleksji historyka nauki

45%

Maślanka K.

Zagadnienia Filozoficzne w Nauce

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2015

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issue 58

115-150

EN

The paper deals with the ever growing role of computers in pure mathematics. Several examples, mainly from number theory, when numerical experiments did shed some light on difficult problems are given.

15

O niektórych aspektach platońskiej filozofii matematyki

45%

Dembiński B.

Zagadnienia Filozoficzne w Nauce

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2015

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issue 58

45-61

EN

Modern philosophers of mathematics in their discussions tend to refer to mathematical Platonism. Usually they believe that they talk about philosophical thought of Plato himself and understanding of mathematics that was introduced by the ancient philosopher. Unfortunately, contemporary mathematical Platonism has very little in common with original Platonism. In this paper I would like to clarify this issue and present Plato’s philosophy of mathematics.

16

Rozumienie dowodu matematycznego a zagadnienie wyjaśnienia w matematyce

45%

Wójtowicz K.

Zagadnienia Filozoficzne w Nauce

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2015

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issue 58

89-114

EN

In the article, I present two possible points of view concerning mathematical proofs: (a) the formal view (according to which the formalized versions of mathematical proofs reveal their “essence”); (b) the semantic view (according to which mathematical proofs are sequences of intellectual acts, and a form of intuitive “grasp” is crucial). The problem of formalizability of mathematical proofs is discussed, as well as the problem of explanation in mathematics – in particular the problem of explanatory versus non-explanatory character of mathematical proofs. I argue, that this problem can be analyzed in a fruitful way only from the semantic point of view.

17

Matematyka - nauka o fikcjach?

44%

Wójtowicz K.

Zagadnienia Filozoficzne w Nauce

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2009

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issue 45

3-26

PL

According to mathematical realism, mathematics describes an abstract realm of mathematical entities, and mathematical theorems are true in the classical sense of this term. In particular, mathematical realism is claimed to be the best theoretical explanation of the applicability of mathematics in science. According to Quine's indispensability argument, applicability is the best argument available in favor of mathematical realism. However, Quine's point of view has been questioned several times by the adherents of antirealism. According to Field, it is possible to show, that - in principle - mathematics is dispensable, and that so called synthetic versions of empirical theories are available. In his 'Science Without Numbers' Field follows the 'geometric strategy' - his aim is to reconstruct standard mathematical techniques in a suitable language, acceptable from the point of view of the nominalist. In the first part of the article, the author briefly presents Field's strategy. The second part is devoted to Balaguer's fictionalism, according to which mathematics is indispensable in science, but nevertheless can be considered to be a merely useful fiction.

18

Dowód matematyczny – argumentacja czy derywacja? – część II

44%

Wójtowicz K.

Zagadnienia Filozoficzne w Nauce

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2011

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issue 49

81-97

PL

In the first part of the paper, Azzouni’s derivation–indicator view was presented. In the second part it is analyzed in a detailed way. It is shown, that many problems arise, which cannot be explained in a satisfactory way in Azzouni’s theory, in particular the problem of the explanatory role of proof, of its epistemic role; the relationship between first–order and second–order versions of proofs is also not clear. It is concluded, that Azzouni’s theory does not provide a satisfactory account of mathematical proof, but inspires an interesting discussion. In the article, some of the mentioned problems are discussed.

19

Dowód matematyczny – argumentacja czy derywacja? – część I

44%

Wójtowicz K.

Zagadnienia Filozoficzne w Nauce

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2011

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issue 49

63-80

PL

The article is devoted to the problem of status of mathematical proofs, in particular it tries to capture the relationship between the real, „semantic” notion of mathematical proof, and its formal (algorithmic) counterpart. In the first part, Azzouni’s derivation–indicator view is presented in a detailed way. According to the DI view, there is a formal derivation underlying every real proof.

20

Matematyka w teologii, teologia w matematyce

44%

Krajewski S.

Zagadnienia Filozoficzne w Nauce

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2016

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issue 60

99-118

PL

Mathematicians use theological metaphors when they talk in the kitchen of mathematics. How essential is this talk? Have theological considerations and religious concepts influenced mathematics? Can mathematical models illuminate theology? Some authors have given positive answers to these questions, but they do not seem final. It is unclear how religious views influenced the work of those mathematicians who were also theologians. Religious background of some mathematical concepts could have been inessential. Mathematical models in theology have no predictive value. It is, however, important to continue the recently initiated search for the mutual influences of mathematics and theology. (In addition to the references listed at the end of this paper, one can also consult the volume “Theology in Mathematics?” ed. by Stanisław Krajewski and Kazimierz Trzęsicki, Studies in Logic, Grammar and Rhetoric 44 (57), 2016.)

Refine search results

Kilka uwag o (meta)filozofii matematyki

What is Diagrammatic Reasoning in Mathematics?

Idea, číslo, pravidlo

U zródeł współczesnego nominalizmu w filozofii matematyki

Aksjomat multiplikatywny Russela

Kilka uwag o Tezie Churcha i Aksjomacie Hilberta

Metoda matematyki według B. Bolzano

Jak přeměnit kvantitu v kvalitu? : K Hegelovu pojmu míry

Kripkenstein from the mathematical point of view: a preliminary survey

Filozofia matematyki Wacława Sierpińskiego

Co nowego w filozoficznym problemie matematyczności przyrody?

Fizyzm Rolanda Omnésa - jedność świata matematyki i fizyki. Część I: kwantowe problemy abstrakcji

Podmiot matematyczny Hilberta

Matematyka eksperymentalna – kilka refleksji historyka nauki

O niektórych aspektach platońskiej filozofii matematyki

Rozumienie dowodu matematycznego a zagadnienie wyjaśnienia w matematyce

Matematyka - nauka o fikcjach?

Dowód matematyczny – argumentacja czy derywacja? – część II

Dowód matematyczny – argumentacja czy derywacja? – część I

Matematyka w teologii, teologia w matematyce